# N-abelian group

This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property

## Definition

Suppose $n$ is an integer. A group $G$ is termed a $n$-abelian group if the $n^{th}$ power map $x \mapsto x^n$ is an endomorphism of $G$, i.e., $(xy)^n = x^ny^n$ for all $x,y \in G$. If this is the case, then the $n^{th}$ power map is termed a universal power endomorphism of $G$.

As noted below, n-abelian iff (1-n)-abelian, so it suffices to restrict attention to $n$ a positive integer.

## Facts

### Particular values

Value of $n$ (note that the condition for $n$ is the same as the condition for $1-n$) Characterization of $n$-abelian groups Proof Other related facts
0 all groups obvious
1 all groups obvious
2 abelian groups only 2-abelian iff abelian endomorphism sends more than three-fourths of elements to squares implies abelian
-1 abelian groups only -1-abelian iff abelian
3 3-abelian group means: 2-Engel group and derived subgroup has exponent dividing three Levi's characterization of 3-abelian groups cube map is surjective endomorphism implies abelian, cube map is endomorphism iff abelian (if order is not a multiple of 3), cube map is endomorphism implies class three
-2 same as for 3-abelian (based on n-abelian iff (1-n)-abelian)